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Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function, ζ(s), is a function of a complex variable s that analytically continues the sum of the Dirichlet series

Riemann zeta function
Riemann-Zeta-Func.png
The Riemann zeta function ζ(z) plotted with domain coloring.[1]
Basic features
Domain
Codomain
 
Specific values
At zero
Limit to +
Value at 
Value at 
Value at 
Value at 
 
 
The pole at , and two zeros on the critical line.

which converges when the real part of s is greater than 1. More general representations of ζ(s) for all s are given below. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.

As a function of a real variable, Leonhard Euler first introduced and studied it in the first half of the eighteenth century without using complex analysis, which was not available at the time. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers.[2]

The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, ζ(2), provides a solution to the Basel problem. In 1979 Apéry proved the irrationality of ζ(3). The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known.

Contents

DefinitionEdit

 
Bernhard Riemann's article On the number of primes below a given magnitude.

The Riemann zeta function ζ(s) is a function of a complex variable s = σ + it. (The notation s, σ, and t is used traditionally in the study of the zeta function, following Riemann.)

The following infinite series converges for all complex numbers s with real part greater than 1, and defines ζ(s) in this case:

 

It can also be defined by the integral

 

where

 

is the gamma function.

The Riemann zeta function is defined as the analytic continuation of the function defined for σ > 1 by the sum of the preceding series.

Leonhard Euler considered the above series in 1740 for positive integer values of s, and later Chebyshev extended the definition to Re(s) > 1.[3]

The above series is a prototypical Dirichlet series that converges absolutely to an analytic function for s such that σ > 1 and diverges for all other values of s. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values s ≠ 1. For s = 1 the series is the harmonic series which diverges to +∞, and

 

Thus the Riemann zeta function is a meromorphic function on the whole complex s-plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1.

Specific valuesEdit

 
Riemann zeta function for three different intervals of  .

For any positive even integer 2n:

 

where B2n is the 2nth Bernoulli number.

For odd positive integers, no such simple expression is known, although these values are thought to be related to the algebraic K-theory of the integers; see Special values of L-functions.

For nonpositive integers, one has

 

for n ≥ 0 (using the NIST convention that B1 = −1/2)

In particular, ζ vanishes at the negative even integers because Bm = 0 for all odd m other than 1.

Via analytic continuation, one can show that:

  •  
This gives a way to assign a finite result to the divergent series 1 + 2 + 3 + 4 + ⋯, which has been used in certain contexts such as string theory.[4]
  •  
Similarly to the above, this assigns a finite result to the series 1 + 1 + 1 + 1 + ⋯.
  •     ( A059750)
This is employed in calculating of kinetic boundary layer problems of linear kinetic equations.[5]
  •  
if we approach from numbers larger than 1. Then this is the harmonic series. But its Cauchy principal value
 
exists which is the Euler–Mascheroni constant γ = 0.5772….
  •     ( A078434)
This is employed in calculating the critical temperature for a Bose–Einstein condensate in a box with periodic boundary conditions, and for spin wave physics in magnetic systems.
  •     ( A013661)
The demonstration of this equality is known as the Basel problem. The reciprocal of this sum answers the question: What is the probability that two numbers selected at random are relatively prime?[6]
  •     ( A002117)
This number is called Apéry's constant.
  •     ( A0013662)
This appears when integrating Planck's law to derive the Stefan–Boltzmann law in physics.

Euler product formulaEdit

The connection between the zeta function and prime numbers was discovered by Euler, who proved the identity

 

where, by definition, the left hand side is ζ(s) and the infinite product on the right hand side extends over all prime numbers p (such expressions are called Euler products):

 

Both sides of the Euler product formula converge for Re(s) > 1. The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic. Since the harmonic series, obtained when s = 1, diverges, Euler's formula (which becomes p p/p − 1) implies that there are infinitely many primes.[7]

The Euler product formula can be used to calculate the asymptotic probability that s randomly selected integers are set-wise coprime. Intuitively, the probability that any single number is divisible by a prime (or any integer) p is 1/p. Hence the probability that s numbers are all divisible by this prime is 1/ps, and the probability that at least one of them is not is 1 − 1/ps. Now, for distinct primes, these divisibility events are mutually independent because the candidate divisors are coprime (a number is divisible by coprime divisors n and m if and only if it is divisible by nm, an event which occurs with probability 1/nm). Thus the asymptotic probability that s numbers are coprime is given by a product over all primes,

 

(More work is required to derive this result formally.)[8]

Riemann's functional equationEdit

The zeta function satisfies the functional equation:

 

where Γ(s) is the gamma function. This is an equality of meromorphic functions valid on the whole complex plane. The equation relates values of the Riemann zeta function at the points s and 1 − s, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that ζ(s) has a simple zero at each even negative integer s = −2n, known as the trivial zeros of ζ(s). When s is an even positive integer, the product sin(πs/2)Γ(1 − s) on the right is non-zero because Γ(1 − s) has a simple pole, which cancels the simple zero of the sine factor.

Proof of functional equation

A proof of the functional equation proceeds as follows: We observe that, if  
 
As a result, if  
 
With the inversion of the limiting processes justified by absolute convergence (hence the stricter requirement on  )
For convenience, let
 
Then  
Given that  
Then  
Hence  
This is equivalent to  
Or  
 
Which is convergent for all s, so holds by analytic continuation. Furthermore, the RHS is unchanged if s is changed to 1-s. Hence
 
Which is the functional equation.
E. C. Titchmarsh (1986). The Theory of the Riemann Zeta-function (2nd ed.). Oxford: Oxford Science Publications. pp. 21–22. ISBN 0-19-853369-1. Attributed to Bernhard Riemann.

The functional equation was established by Riemann in his 1859 paper "On the Number of Primes Less Than a Given Magnitude" and used to construct the analytic continuation in the first place. An equivalent relationship had been conjectured by Euler over a hundred years earlier, in 1749, for the Dirichlet eta function (alternating zeta function):

 

Incidentally, this relation gives an equation for calculating ζ(s) in the region 0 < Re(s) < 1, i.e.

 

where the η-series is convergent (albeit non-absolutely) in the larger half-plane s > 0 (for a more detailed survey on the history of the functional equation, see e.g. Blagouchine[9][10]).

Riemann also found a symmetric version of the functional equation applying to the xi-function:

 

which satisfies:

 

(Riemann's original ξ(t) was slightly different.)

Zeros, the critical line, and the Riemann hypothesisEdit

 
Apart from the trivial zeros, the Riemann zeta function has no zeros to the right of σ = 1 and to the left of σ = 0 (neither can the zeros lie too close to those lines). Furthermore, the non-trivial zeros are symmetric about the real axis and the line σ = 1/2 and, according to the Riemann hypothesis, they all lie on the line σ = 1/2.
 
This image shows a plot of the Riemann zeta function along the critical line for real values of t running from 0 to 34. The first five zeros in the critical strip are clearly visible as the place where the spirals pass through the origin.
 
The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non-trivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011.

The functional equation shows that the Riemann zeta function has zeros at −2, −4,…. These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from sin πs/2 being 0 in the functional equation. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields impressive results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip {s : 0 < Re(s) < 1}, which is called the critical strip. The Riemann hypothesis, considered one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero s has Re(s) = 1/2. In the theory of the Riemann zeta function, the set {s : Re(s) = 1/2} is called the critical line. For the Riemann zeta function on the critical line, see Z-function.

The Hardy–Littlewood conjecturesEdit

In 1914, Godfrey Harold Hardy proved that ζ (1/2 + it) has infinitely many real zeros.

Hardy and John Edensor Littlewood formulated two conjectures on the density and distance between the zeros of ζ (1/2 + it) on intervals of large positive real numbers. In the following, N(T) is the total number of real zeros and N0(T) the total number of zeros of odd order of the function ζ (1/2 + it) lying in the interval (0, T].

  1. For any ε > 0, there exists a T0(ε) > 0 such that when
     
    the interval (T, T + H] contains a zero of odd order.
  2. For any ε > 0, there exists a T0(ε) > 0 and cε > 0 such that the inequality
     
    holds when
     .

These two conjectures opened up new directions in the investigation of the Riemann zeta function.

Zero-free regionEdit

The location of the Riemann zeta function's zeros is of great importance in the theory of numbers. The prime number theorem is equivalent to the fact that there are no zeros of the zeta function on the Re(s) = 1 line.[11] A better result[12] that follows from an effective form of Vinogradov's mean-value theorem is that ζ (σ + it) ≠ 0 whenever |t| ≥ 3 and

 

The strongest result of this kind one can hope for is the truth of the Riemann hypothesis, which would have many profound consequences in the theory of numbers.

Other resultsEdit

It is known that there are infinitely many zeros on the critical line. Littlewood showed that if the sequence (γn) contains the imaginary parts of all zeros in the upper half-plane in ascending order, then

 

The critical line theorem asserts that a positive proportion of the nontrivial zeros lies on the critical line. (The Riemann hypothesis would imply that this proportion is 1.)

In the critical strip, the zero with smallest non-negative imaginary part is 1/2 + 14.13472514…i ( A058303). The fact that

 

for all complex s ≠ 1 implies that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line Re(s) = 1/2.

Various propertiesEdit

For sums involving the zeta-function at integer and half-integer values, see rational zeta series.

ReciprocalEdit

The reciprocal of the zeta function may be expressed as a Dirichlet series over the Möbius function μ(n):

 

for every complex number s with real part greater than 1. There are a number of similar relations involving various well-known multiplicative functions; these are given in the article on the Dirichlet series.

The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of s is greater than 1/2.

UniversalityEdit

The critical strip of the Riemann zeta function has the remarkable property of universality. This zeta-function universality states that there exists some location on the critical strip that approximates any holomorphic function arbitrarily well. Since holomorphic functions are very general, this property is quite remarkable. The first proof of universality was provided by Sergei Mikhailovitch Voronin in 1975.[13] More recent work has included effective versions of Voronin's theorem[14] and extending it to Dirichlet L-functions.[15][16]

Estimates of the maximum of the modulus of the zeta functionEdit

Let the functions F(T;H) and G(s0;Δ) be defined by the equalities

 

Here T is a sufficiently large positive number, 0 < H ≪ ln ln T, s0 = σ0 + iT, 1/2σ0 ≤ 1, 0 < Δ < 1/3. Estimating the values F and G from below shows, how large (in modulus) values ζ(s) can take on short intervals of the critical line or in small neighborhoods of points lying in the critical strip 0 ≤ Re(s) ≤ 1.

The case H ≫ ln ln T was studied by Kanakanahalli Ramachandra; the case Δ > c, where c is a sufficiently large constant, is trivial.

Anatolii Karatsuba proved,[17][18] in particular, that if the values H and Δ exceed certain sufficiently small constants, then the estimates

 

hold, where c1 and c2 are certain absolute constants.

The argument of the Riemann zeta functionEdit

The function

 

is called the argument of the Riemann zeta function. Here arg ζ(1/2 + it) is the increment of an arbitrary continuous branch of arg ζ(s) along the broken line joining the points 2, 2 + it and 1/2 + it.

There are some theorems on properties of the function S(t). Among those results[19][20] are the mean value theorems for S(t) and its first integral

 

on intervals of the real line, and also the theorem claiming that every interval (T, T + H] for

 

contains at least

 

points where the function S(t) changes sign. Earlier similar results were obtained by Atle Selberg for the case

 .

RepresentationsEdit

Dirichlet seriesEdit

An extension of the area of convergence can be obtained by rearranging the original series.[21] The series

 

converges for Re(s) > 0, while

 

converges even for Re(s) > −1. In this way, the area of convergence can be extended to Re(s) > −k for any negative integer k.

Mellin-type integralsEdit

The Mellin transform of a function f(x) is defined as

 

in the region where the integral is defined. There are various expressions for the zeta-function as Mellin transform-like integrals. If the real part of s is greater than one, we have

 

where Γ denotes the gamma function. By modifying the contour, Riemann showed that

 

for all s (where H denotes the Hankel contour).

Starting with the integral formula   one can show[22] by substitution and iterated differentation for natural  

 

using the notation of umbral calculus where each power   is to be replaced by  , so e.g. for   we have   while for   this becomes

 

We can also find expressions which relate to prime numbers and the prime number theorem. If π(x) is the prime-counting function, then

 

for values with Re(s) > 1.

A similar Mellin transform involves the Riemann prime-counting function J(x), which counts prime powers pn with a weight of 1/n, so that

 

Now we have

 

These expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. Riemann's prime-counting function is easier to work with, and π(x) can be recovered from it by Möbius inversion.

Theta functionsEdit

The Riemann zeta function can be given by a Mellin transform[23]

 

in terms of Jacobi's theta function

 

However, this integral only converges if the real part of s is greater than 1, but it can be regularized. This gives the following expression for the zeta function, which is well defined for all s except 0 and 1:

 

Laurent seriesEdit

The Riemann zeta function is meromorphic with a single pole of order one at s = 1. It can therefore be expanded as a Laurent series about s = 1; the series development is then

 

The constants γn here are called the Stieltjes constants and can be defined by the limit

 

The constant term γ0 is the Euler–Mascheroni constant.

IntegralEdit

For all s, s ≠ 1 the integral relation (cf. Abel–Plana formula)

 

holds true, which may be used for a numerical evaluation of the zeta-function.

Rising factorialEdit

Another series development using the rising factorial valid for the entire complex plane is[citation needed]

 

This can be used recursively to extend the Dirichlet series definition to all complex numbers.

The Riemann zeta function also appears in a form similar to the Mellin transform in an integral over the Gauss–Kuzmin–Wirsing operator acting on xs − 1; that context gives rise to a series expansion in terms of the falling factorial.[24]

Hadamard productEdit

On the basis of Weierstrass's factorization theorem, Hadamard gave the infinite product expansion

 

where the product is over the non-trivial zeros ρ of ζ and the letter γ again denotes the Euler–Mascheroni constant. A simpler infinite product expansion is

 

This form clearly displays the simple pole at s = 1, the trivial zeros at −2, −4, … due to the gamma function term in the denominator, and the non-trivial zeros at s = ρ. (To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeros, i.e. the factors for a pair of zeros of the form ρ and 1 − ρ should be combined.)

Globally convergent seriesEdit

A globally convergent series for the zeta function, valid for all complex numbers s except s = 1 + i/ln 2n for some integer n, was conjectured by Konrad Knopp[25] and proven by Helmut Hasse in 1930[26] (cf. Euler summation):

 

The series only appeared in an appendix to Hasse's paper, and did not become generally known until it was discussed by Jonathan Sondow in 1994.[27]

Hasse also proved the globally converging series

 

in the same publication,[26] but research by Iaroslav Blagouchine[28][25] has found that this latter series was actually first published by Joseph Ser in 1926.[29] New proofs for both of these results were offered by Demetrios Kanoussis in 2017.[30] Other similar globally convergent series include

 

where Hn are the Harmonic numbers,   are the Stirling numbers of the first kind,   is the Pochhammer symbol, Gn are the Gregory coefficients, G(k)
n
are the Gregory coefficients of higher order, Cn are the Cauchy numbers of the second kind (C1 = 1/2, C2 = 5/12, C3 = 3/8,...), and ψn(a) are the Bernoulli polynomials of the second kind, see Blagouchine's paper.[25]

Peter Borwein has developed an algorithm that applies Chebyshev polynomials to the Dirichlet eta function to produce a very rapidly convergent series suitable for high precision numerical calculations.[31]

Series representation at positive integers via the primorialEdit

 

Here pn# is the primorial sequence and Jk is Jordan's totient function.[32]

Series representation by the incomplete poly-Bernoulli numbersEdit

The function ζ can be represented, for Re(s) > 1, by the infinite series

 

where k ∈ {−1, 0}, Wk is the kth branch of the Lambert W-function, and B(μ)
n, ≥2
is an incomplete poly-Bernoulli number.[33]

The Mellin transform of the Engel mapEdit

The function :  is iterated to find the coeffecients appearing in Engel expansions.

The Mellin transform of the map   is related to the Riemann zeta function by the formula

 

Numerical algorithmsEdit

For   , the Riemann zeta function has for fixed   and for all   the following representation in terms of three absolutely and uniformly converging series,[34]

 
where for positive integer   one has to take the limit value  . The derivatives of   can be calculated by differentiating the above series termwise. From this follows an algorithm which allows to compute, to arbitrary precision,   and its derivatives using at most   summands for any  , with explicit error bounds. For  , these are as follows:

For a given argument   with   and   one can approximate   to any accuracy   by summing the first series to  ,   to   and neglecting  , if one chooses   as the next higher integer of the unique solution of   in the unknown  , and from this  . For   one can neglect   altogether. Under the mild condition   one needs at most   summands. Hence this algorithm is essentially as fast as the Riemann-Siegel formula. Similar algorithms are possible for Dirichlet L-functions.[34]

ApplicationsEdit

The zeta function occurs in applied statistics (see Zipf's law and Zipf–Mandelbrot law).

Zeta function regularization is used as one possible means of regularization of divergent series and divergent integrals in quantum field theory. In one notable example, the Riemann zeta-function shows up explicitly in one method of calculating the Casimir effect. The zeta function is also useful for the analysis of dynamical systems.[35]

Infinite seriesEdit

The zeta function evaluated at equidistant positive integers appears in infinite series representations of a number of constants.[36]

  •  

In fact the even and odd terms give the two sums

  •  

and

  •  

Parametrized versions of the above sums are given by

  •  

and

  •  

with   and where   and   are the Polygamma function and Euler's constant, as well as

  •  

all of which are continuous at  . Other sums include

  •  
  •  
  •  
  •  

where Im denotes the imaginary part of a complex number.

There are yet more formulas in the article Harmonic number.

GeneralizationsEdit

There are a number of related zeta functions that can be considered to be generalizations of the Riemann zeta function. These include the Hurwitz zeta function

 

(the convergent series representation was given by Helmut Hasse in 1930,[26] cf. Hurwitz zeta function), which coincides with the Riemann zeta function when q = 1 (note that the lower limit of summation in the Hurwitz zeta function is 0, not 1), the Dirichlet L-functions and the Dedekind zeta-function. For other related functions see the articles zeta function and L-function.

The polylogarithm is given by

 

which coincides with the Riemann zeta function when z = 1.

The Lerch transcendent is given by

 

which coincides with the Riemann zeta function when z = 1 and q = 1 (note that the lower limit of summation in the Lerch transcendent is 0, not 1).

The Clausen function Cls(θ) that can be chosen as the real or imaginary part of Lis(e).

The multiple zeta functions are defined by

 

One can analytically continue these functions to the n-dimensional complex space. The special values of these functions are called multiple zeta values by number theorists and have been connected to many different branches in mathematics and physics.

Fractional derivativeEdit

In the case of the Riemann zeta function, a difficulty is represented by the fractional differentiation in the complex plane. The Ortigueira generalization of the classical Caputo fractional derivative solves this problem. The  -order fractional derivative of the Riemann zeta function is given by [37]

 

Given that   is a fractional number such that  , the half-plane of convergence is  .

See alsoEdit

NotesEdit

  1. ^ "Jupyter Notebook Viewer". Nbviewer.ipython.org. Retrieved 2017-01-04.
  2. ^ This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that is considered by many mathematicians to be the most important unsolved problem in pure mathematics.Bombieri, Enrico. "The Riemann Hypothesis – official problem description" (PDF). Clay Mathematics Institute. Retrieved 2014-08-08.
  3. ^ Devlin, Keith (2002). The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time. New York: Barnes & Noble. pp. 43–47. ISBN 978-0-7607-8659-8.
  4. ^ Polchinski, Joseph (1998). String Theory, Volume I: An Introduction to the Bosonic String. Cambridge University Press. p. 22. ISBN 978-0-521-63303-1.
  5. ^ Kainz, A. J.; Titulaer, U. M. (1992). "An accurate two-stream moment method for kinetic boundary layer problems of linear kinetic equations". J. Phys. A: Math. Gen. 25 (7): 1855–1874. Bibcode:1992JPhA...25.1855K. doi:10.1088/0305-4470/25/7/026.
  6. ^ Ogilvy, C. S.; Anderson, J. T. (1988). Excursions in Number Theory. Dover Publications. pp. 29–35. ISBN 0-486-25778-9.
  7. ^ Sandifer, Charles Edward (2007). How Euler Did It. Mathematical Association of America. p. 193. ISBN 978-0-88385-563-8.
  8. ^ Nymann, J. E. (1972). "On the probability that k positive integers are relatively prime". Journal of Number Theory. 4 (5): 469–473. Bibcode:1972JNT.....4..469N. doi:10.1016/0022-314X(72)90038-8.
  9. ^ I. V. Blagouchine The history of the functional equation of the zeta-function. Seminar on the History of Mathematics, Steklov Institute of Mathematics at St. Petersburg, 1 March 2018. PDF
  10. ^ I. V. Blagouchine Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results. The Ramanujan Journal, vol. 35, no. 1, pp. 21-110, 2014. Addendum: vol. 42, pp. 777-781, 2017. PDF
  11. ^ Diamond, Harold G. (1982). "Elementary methods in the study of the distribution of prime numbers". Bulletin of the American Mathematical Society. 7 (3): 553–89. doi:10.1090/S0273-0979-1982-15057-1. MR 0670132.
  12. ^ Ford, K. (2002). "Vinogradov's integral and bounds for the Riemann zeta function". Proc. London Math. Soc. 85 (3): 565–633. doi:10.1112/S0024611502013655.
  13. ^ Voronin, S. M. (1975). "Theorem on the Universality of the Riemann Zeta Function". Izv. Akad. Nauk SSSR, Ser. Matem. 39: 475–486. Reprinted in Math. USSR Izv. (1975) 9: 443–445.
  14. ^ Ramūnas Garunkštis; Antanas Laurinčikas; Kohji Matsumoto; Jörn Steuding; Rasa Steuding (2010). "Effective uniform approximation by the Riemann zeta-function". Publicacions Matemàtiques. 54: 209–219. doi:10.1090/S0025-5718-1975-0384673-1. JSTOR 43736941.
  15. ^ Bhaskar Bagchi (1982). "A Joint Universality Theorem for Dirichlet L-Functions". Mathematische Zeitschrift. 181: 319–334. doi:10.1007/bf01161980. ISSN 0025-5874.
  16. ^ Steuding, Jörn (2007). Value-Distribution of L-Functions. Lecture Notes in Mathematics. Berlin: Springer. p. 19. doi:10.1007/978-3-540-44822-8. ISBN 3-540-26526-0.
  17. ^ Karatsuba, A. A. (2001). "Lower bounds for the maximum modulus of ζ(s) in small domains of the critical strip". Mat. Zametki. 70 (5): 796–798.
  18. ^ Karatsuba, A. A. (2004). "Lower bounds for the maximum modulus of the Riemann zeta function on short segments of the critical line". Izv. Ross. Akad. Nauk, Ser. Mat. 68 (8): 99–104.
  19. ^ Karatsuba, A. A. (1996). "Density theorem and the behavior of the argument of the Riemann zeta function". Mat. Zametki (60): 448–449.
  20. ^ Karatsuba, A. A. (1996). "On the function S(t)". Izv. Ross. Akad. Nauk, Ser. Mat. 60 (5): 27–56.
  21. ^ Knopp, Konrad (1945). Theory of Functions. pp. 51–55.
  22. ^ "Evaluating the definite integral..." math.stackexchange.com.
  23. ^ Neukirch, Jürgen (1999). Algebraic number theory. Springer. p. 422. ISBN 3-540-65399-6.
  24. ^ "A series representation for the Riemann Zeta derived from the Gauss-Kuzmin-Wirsing Operator" (PDF). Linas.org. Retrieved 2017-01-04.
  25. ^ a b c Blagouchine, Iaroslav V. (2018). "Three Notes on Ser's and Hasse's Representations for the Zeta-functions". Integers (Electronic Journal of Combinatorial Number Theory). 18A: 1–45. arXiv:1606.02044. Bibcode:2016arXiv160602044B.
  26. ^ a b c Hasse, Helmut (1930). "Ein Summierungsverfahren für die Riemannsche ζ-Reihe" [A summation method for the Riemann ζ series]. Mathematische Zeitschrift (in German). 32 (1): 458–464. doi:10.1007/BF01194645.
  27. ^ Sondow, Jonathan (1994). "Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series" (PDF). Proceedings of the American Mathematical Society. 120 (2): 421–424. doi:10.1090/S0002-9939-1994-1172954-7.
  28. ^ Blagouchine, Iaroslav V. (2016). "Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only". Journal of Number Theory. 158: 365–396. arXiv:1501.00740. doi:10.1016/j.jnt.2015.06.012.
  29. ^ Ser, Joseph (1926). "Sur une expression de la fonction ζ(s) de Riemann" [Upon an expression for Riemann's ζ function]. Comptes rendus hebdomadaires des séances de l'Académie des Sciences (in French). 182: 1075–1077.
  30. ^ Kanoussis, Demetrios P. (2017). "A New Proof of H. Hasse's Global Expression for the Riemann's Zeta Function".
  31. ^ Borwein, Peter (2000). "An Efficient Algorithm for the Riemann Zeta Function". In Théra, Michel A. Constructive, Experimental, and Nonlinear Analysis (PDF). Conference Proceedings, Canadian Mathematical Society. 27. Providence, RI: American Mathematical Society, on behalf of the Canadian Mathematical Society. pp. 29–34. ISBN 978-0-8218-2167-1.
  32. ^ Mező, István (2013). "The primorial and the Riemann zeta function". The American Mathematical Monthly. 120 (4): 321.
  33. ^ Komatsu, Takao; Mező, István (2016). "Incomplete poly-Bernoulli numbers associated with incomplete Stirling numbers". Publicationes Mathematicae Debrecen. 88 (3–4): 357–368. arXiv:1510.05799. doi:10.5486/pmd.2016.7361.
  34. ^ a b Fischer, Kurt (2017-03-04). "The Zetafast algorithm for computing zeta functions". arXiv:1703.01414.
  35. ^ "Work on spin-chains by A. Knauf, et. al". Empslocal.ex.ac.uk. Retrieved 2017-01-04.
  36. ^ Most of the formulas in this section are from § 4 of J. M. Borwein et al. (2000)
  37. ^ Guariglia, E. (2015). Fractional derivative of the Riemann zeta function. In: Fractional Dynamics (Cattani, C., Srivastava, H., and Yang, X. Y.). De Gruyter. pp. 357–368. doi:10.1515/9783110472097-022.

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